Hopf algebras, tetramodules, and n-fold monoidal categories

TL;DR

This paper explores the structure of tetramodules over bialgebras, establishing a connection between their Ext groups and higher algebraic structures, and demonstrates that Gerstenhaber-Schack cohomology forms a 3-algebra for Hopf algebras.

Contribution

It constructs a 2-fold monoidal structure on tetramodules and proves that Ext groups in n-fold monoidal categories form (n+1)-algebras under certain conditions, notably for Hopf algebras.

Findings

Gerstenhaber-Schack cohomology of a Hopf algebra is a 3-algebra.

A 2-fold monoidal structure exists on the category of tetramodules over a bialgebra.

The operad acting on Hochschild cohomology over integers is the stable homotopy groups of the little discs operad.

Abstract

The abelian category of tetramodules over an associative bialgebra $A$ is related with the Gerstenhaber-Schack (GS) cohomology as $Ext_\Tetra(A,A)=H_\GS(A)$. We construct a 2-fold monoidal structure on the category of tetramodules of a bialgebra. Suppose $C$ is an abelian $n$-fold monoidal category with the unit object $A$. We prove, provided some condition (*), that $Ext_C(A,A)$ is an $(n+1)$-algebra. In the case of bialgebras this condition (*) is satisfied when $A$ is a Hopf algebra. Finally, the GS cohomology of a Hopf algebra is a 3-algebra. As well, we consider this kind of questions of (bi)algebras over integers. Let $A$ be an associative algebra over $Z$ flat over $Z$. We prove that the operad acting on its Hochschild cohomology is the operad of stable homotopy groups of the little discs operad.